Test 1

Libraries Required

library(rgl)

## This build of rgl does not include OpenGL functions.  Use
##  rglwidget() to display results, e.g. via options(rgl.printRglwidget = TRUE).

options(rgl.printRglwidget = TRUE)

Visualization function

visualize3D <- function(fname ,range){
  
  x<- y<- seq(-range,range,length = 100)
  data<-data.matrix(expand.grid(x,y))

  
  `f(x,y)`<-matrix(apply(data,1,fname),
             nrow = length(x),
             ncol = length(y))
  
  div<-100
  color <- rev(rainbow(div))
  zcol <- cut(`f(x,y)`, div)
  
  persp3d(x,y,`f(x,y)` , col = color[zcol])
}

1) Ackley Function

1.1) Equation of Ackley function

\[ f(x) = -a\,exp \left(-b\,\sqrt{\frac{1}{d} \sum_{i=1}^{d}{x_i^2}}\right)- \,exp\left(-b\,{\frac{1}{d} \sum_{i=1}^{d}{cos(cx_i)}}\right) + \,a+\,exp(1) \]

1.2) R implementation of Ackley function

ackley<-function(x,a=20, b=0.2 , c =2*pi){
  d <- length(x)
  
  sum1 <- sum(x^2)
  sum2 <- sum(cos(2*pi*x))
  
  term1 <- -a * exp(-b*sqrt(sum1/d))
  term2 <- -exp(sum2/d)
  
  y <- term1 + term2 + a + exp(1)
  return(y)
}

1.3) Ackley function visualization

visualize3D(ackley,32)

2) Zakharov Function

2.1) Equation of Zakharov Function

\[ Z_n(x) = \sum_{i=1}^{n}{x_i^2} + \left(\sum_{i=1}^{n}{0.5ix_i}\right)^2 + \left(\sum_{i=1}^{n}{0.5ix_i}\right)^4 \]

2.2) R implementation of Zakharov function

zakharov <- function(x){
  
  term1 <-sum(x^2)
  i <-seq_along(x)
  term2 <- sum(i*0.5*(x))
  
  y <- term1 + term2^2 + term2^4
  return(y)
}

2.3) Visulization of Zakharov function

visualize3D(zakharov,10)

3) Dixon-price function

3.1) Dixon-price function equation

\[ f(x) = (x_1 - 1)^2 +\sum_{i=2}^{n}{i(2x_i^2-x_{i-1})^2} \]

3.2) R implementation of Dixon-price function

dixon <- function(x){
  
  term1 <-(x[1]-1)^2
  
  # computation of sum starts from i = 2 thus taking lag befor removing values
  d <- length(x)-1
  xlag <- x[1:d]
  
  x <- x[-1]
  i <- seq_along(x)+1
  
  term2 <- sum(i*(2*x^2 - xlag)^2 ) 
  
  y <- term1 + term2
  
  return(y)
  
}

3.3) Dixon-price function visulization

visualize3D(dixon,10)

4) Griewank function

4.1) Griewank function equation

\[ f(x) = \sum_{i=n}{n}\frac{x_i^2}{4000}- \prod_{i=1}^{n}cos\left(\frac{x_i}{\sqrt{i}}\right)+1 \]

4.2) R implementation of Griewank function

griewank <- function(x,y){
  term1 <- sum(x^2)
  term2 <- prod(cos(x)/seq_along(x))
  y <- 1 + (1/4000 * term1) - term2
  
  return(y)
}

4.3) Visulization of Griewank function

visualize3D(griewank,100)

5) Michalewicz Function

5.1) Michalewicz Function equation

\[ f(x) = - \sum_{i=1}^{d}sin(x_i){sin^{2m}\left(\frac{ix_i^2}{\pi}\right)} \]

5.2) R implementation of Griewank function

michale <- function(x,m=10){
  
  d <- length(x)
  i <- seq_along(x)
  
  sum <- sum(sin(x) * (sin(i*x^2/pi))^(2*m))
  
  y <- sum
  return(-y)
}

# new
visualize3D(michale,4)

6) Perm \(P_{n,\beta}\) Function

6.1) Equation of Perm \(P_{n,\beta}\) Function

\[ f(x)=\sum_{i=1}^{d}{ \left(\sum_{j=1}^{d}{(j^i+\beta)} \left({\left({\frac{x_j}{j}}\right)^i}-1\right) \right)^2} \]

6.2) R implementation of Perm \(P_{n,\beta}\) Function

perm <- function(x, b=0.5){
  
  d <- length(x)
  ii <- c(1:d)
  jj <- matrix(rep(ii,times=d), d, d, byrow=TRUE)
  
  xxmat <- matrix(rep(x,times=d), d, d, byrow=TRUE)
  inner <- rowSums((jj^ii+b)*((xxmat/jj)^ii-1)) 
  outer <- sum(inner^2)
  
  y <- outer
  return(y)
}

6.3) Visulization of Perm \(P_{n,\beta}\) Function

visualize3D(perm , 2)

7) Rastrigin Function

7.1) Equation of Rastrigin Function

\[ f(x) = 10d + \sum_{i=1}^{d}[x_i^2 -10cos(2\pi x_i)] \]

7.2) R implementatin of Rastrigin Function

rastrigin <- function(x) {
  y <- 10 * length(x) + sum(x^2 - 10 * cos(2 * pi * x))
  return(y)
}

7.3) Visulization of of Rastrigin Function

visualize3D(rastrigin,5)